The mass flowmeters known by the name of "Convective inertia force" flowmeter of "Coriolis force" flowmeters, which employ a single or a pair of vibrating conduits, have become very popular among many industrial users, as such mass flowmeters provide a number of advantages for the users, which advantages includes the clean flow passage free of any obstruction, the ability to measure mass flow rates of all kinds of media independent of the viscosity and homogeneity of the media and the high accuracy and high repeatability of the mass flow measurement. On the other hand, the mass flowmeters employing the vibrating conduits available at the present time have a few serious short-comings such as the poor sensitivity seriously limiting the ability to measure mass flow rates of low magnitude, and the high cost that prevents the truly wide spread use of the mass flowmeters employing the vibrating conduits. Of course, the high cost of these mass flowmeters is a direct manifestation of the poor sensitivity of the mass flowmeters inherent to the existing designs thereof. The convective inertia force generated by media moving through the vibrating conduits of the mass flowmeter is proportional to the mass flow rate of the media times the gradient in the distribution of the flexural vibratory motion of the conduit following the length of the conduit. The mass flow rate is determined by measuring the amplitude of the secondary flexural vibration of the conduit created by the convective inertia force that is generated by the combination of the mass flow of media through the conduit and the primary flexural vibration of the conduit imposed by an electromagnetic vibrator. Therefore, the sensitivity of the inertia force mass flowmeter can be drastically increased when the vibrating conduit constituting the flow passage in the mass flowmeter has a structural design, that provides, firstly, a sharp curvature in the bending mode of the conduit in the primary flexural vibration and secondly, a low stiffness against the secondary flexural vibration. In more plain language, the inertia force mass flowmeter must employ a conduit or conduits having one or more localized section with high flexibility.